Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 42\cdot 43 + 42\cdot 43^{2} + 9\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 31 + \left(7 a + 12\right)\cdot 43 + \left(9 a + 41\right)\cdot 43^{2} + \left(11 a + 1\right)\cdot 43^{3} + \left(8 a + 24\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 15 + \left(23 a + 20\right)\cdot 43 + \left(21 a + 21\right)\cdot 43^{2} + \left(8 a + 13\right)\cdot 43^{3} + \left(40 a + 35\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 28 + \left(35 a + 23\right)\cdot 43 + \left(33 a + 42\right)\cdot 43^{2} + \left(31 a + 3\right)\cdot 43^{3} + \left(34 a + 21\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 8\cdot 43 + 4\cdot 43^{2} + 13\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 37 + \left(19 a + 21\right)\cdot 43 + \left(21 a + 19\right)\cdot 43^{2} + 34 a\cdot 43^{3} + \left(2 a + 24\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $-1$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
